Question

if x-1/x = 3 find x²+1/x² and x⁴+1/x⁴​

Answer 1

Answer:

9 and 81

Step-by-step explanation:

(x)2+1 / (x)2 = 9.

(x)4+1 / (x)4 = 81

Answer 2

Correct Question :

$\sf \small If \: x - \frac{1}{x} = 3. \: find \: {x}^{2} + \frac{1}{ {x}^{2} } \: an d\: {x}^{4} + \frac{1}{ {x}^{4} }$

To find :

$\sf \small {x}^{2} + \frac{1}{ {x}^{2} } \: an d\: {x}^{4} + \frac{1}{ {x}^{4} }$

Formula Used :

$\sf \small(a - b)^{2} = {a}^{2} -2ab + {b}^{2}$

$\sf \small ({a}^{2} + {b}^{2} )^{2} = {a}^{4} + {b}^{4} +2 {a}^{2} {b}^{2}$

Solution :

So the value of x -1/x i.e. 3 is already given

So the value of x -1/x i.e. 3 is already given It would be helpful for getting x² + 1/x²

$\sf \small i) \: {x}^{2} + \frac{1}{ {x}^{2} }$

$\sf \small \rightarrow (x - \frac{1}{x} )^{2} = {x}^{2} + \frac{1}{ {x}^{2} } -2$

$\sf \small \rightarrow {(3)}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } -2$

$\sf \small \rightarrow9 + 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\boxed{\sf \small \rightarrow \green{ {x}^{2} + \frac{1}{ {x}^{2} } = 11}}$

$\sf \small So \: the \: value \: of \: {x}^{2} + \frac{1}{ {x}^{2} } \:i s \: 11$

Now the second part :

$\sf \small \: ii) \: {x}^{4} + \frac{1}{ {x}^{4} }$

Applying the value of part (I) we'll get the value of the (ii) part

$\sf\small\rightarrow( {x}^{2} + \frac{1}{ {x}^{2} } )^{2} = {x}^{4} + \frac{1}{ {x}^{4} } + 2$

$\sf\small\rightarrow {(11)}^{2} = {x}^{4} + \frac{1}{ {x}^{4} } + 2$

$\sf\small\rightarrow {x}^{4} + \frac{1}{ {x}^{4} } = 121 - 2$

$\boxed{\sf\small\rightarrow \pink{{x}^{4} + \frac{1}{ {x}^{4} } = 119}}$

Final Answer :

$\bullet \: \sf\small Value \: of \: x² + \frac{1}{ {x}^{2} } is \: \boxed{ \blue{11}}$

$\bullet \: \sf\small Value \: of \: x⁴ + \frac{1}{ {x}^{4} } \: is \: \boxed{ \orange{119}}$